Pergamon Stud. Hist. Phil. Mod. Phys., Vol. 28, No. 3, pp. 307-323, 1997 @ 1997 Elsevier Science Ltd. All rights reserved Printed in Great Britain 1355-2198/97 $17.00 + .OO Gauge Theory, Anomalies and Global Geometry: The Interplay of Physics And Mathematics Dana Fine f and Arthur Fine * 1. Introduction Whether today’s physics will lead to a grand theory of everything or to a more modest layer cake of effective theories of different orders (or perhaps to something else), certainly the next theoretical turnings will evolve out of the successes and failures of quantum field theory. In its postwar re-incarnation, field theory has been at the centre of developments not only in the physics of fundamental particles (quantum electrodynamics, chromodynamics and the standard model), but also in condensed-matter physics and cosmology. One of the most striking aspects of field theory in this period has been its alliance with the frontiers of research in ‘pure’ mathematics, especially in the areas of differential geometry and topology. At least since the advent of relativity, the prominent role of abstract mathematics, although not well understood (witness Wigner’s (1967, p. 222) ‘unreasonable effectiveness of mathematics’), has nevertheless been accepted as a fact of life in theoretical physics. What is perhaps new in the context of field theory is the reciprocal feedback of physics on pure mathematics, not just in the promotion of mathematical work in fields and topics allied to physics but also in concrete suggestions, deriving from the physics, for the solution to outstanding mathematical problems. The present work is a small contribution to tracing some of the interplay of physics and mathematics in field theory. Our particular subject is the treatment of anomalies in gauge theory, specifically the global chiral (i.e. left-right) anomalies. In general, the term ‘anomaly’ signifies the breakdown, upon quantisation, of a symmetry present in the classical action where that breakdown does not depend on special features of the vacuum state (as it does in the case of spontaneously broken symmetry). (Received 12 August 1996; revised 2 January 1997) * Department of Philosophy, Northwestern University, Evanston, Illinois 60208, U.S.A. t Department of Mathematics, University of Massachusetts Dartmouth, North Dartmouth, MA 02747, U.S.A. PII:S1355-2198(97)00011-7 307 308 Studies in History and Philosophy of Modern Physics Since symmetries are associated with conservation laws, one can regard the question of anomalies as a question of what remedy to apply when a classical conservation law seems to fail, or actually does fail, in the quantum treatment of the phenomena. A standard example of the former would be the seeming failure of energy conservation in beta decay, whose remedy was the neutrino hypothesis. The chiral anomaly is an example of the latter; that is, chirality is a classical symmetry that actually disappears after (second) quantisation. Originally the chiral anomaly arose in perturbative calculations relating to models of neutral pion dissociation. Using Feynman rules, the calculations were seen to involve a fermion triangle diagram with one axial and two vector currents. Imposing current conservation and Bose symmetry in the vector channels leads to non- conservation of the axial current. This breaks chiral symmetry and results in neutral pion decay Our interest is in showing how this chapter in physics and perturbation theory connects with the introduction of global mathematical methods, and the ramifications of that. The story begins in the laboratory 2. Penetrating Showers in Lead This is the title of a paper by W. B. Fretter (1948) that provides perhaps the first experimental evidence for the existence of neutral mesons. Fretter’s experiment involves a cloud chamber containing eight In-inch thick lead plates, eight inches wide, used to track the production of penetrating showers by cosmic rays. Discussing the interpretation of these showers, Fretter refers to J. R. Oppenheimer, who appears to have been the first to propose the y-instability of neutral mesons when coupled to nucleons: ‘Recently J. R. Oppenheimer has suggested that in these nuclear events not only charged mesons may be produced, but also uncharged mesons. The neutral mesons are calculated to have an extremely short life ( lo-t5 s). Thus, even if the neutral mesons were given a large amount of energy in the initial event they would decay almost immediately and produce a pair of y-rays, which would then create the electron shower’ (pp. 45-46). If it had mattered, Fretter could have improved on the estimate of lo-l5 s, for Oppenheimer had already assigned the problem of calculating the lifetime to R. J. Finkelstein. Finkelstein’s result (1947) gave an order of magnitude estimate of 1 x 1O-t6 s for the decay of neutral ‘pseudo-scalar’ mesons into two photons. This turns out to be in good agreement with the experimental decay rate of (1.12 -t 0.202) x lOi s-l derived from subsequent measurements. 1 The physical picture employed by Finkelstein was that of ‘dressed’ nucleons; i.e. of nucleons surrounded by a virtual meson cloud. Coupling a neutral meson to the electromagnetic field permits y-decay: the neutral pion dissociates into a proton-antiproton pair which then radiates the photons. Finkelstein’s calculations, however, encountered divergent integrals of a sort reminiscent of the self-energy divergences in electrodynamics. Finkelstein’s resolution was to employ a rather ad hoc method of subtraction for cancelling the divergences. 1 For example, see Rosenfeld (1968). Gauge Theory, Anomalies and Global Geometry 309 Fig. 1. The triangle diagram. 3. Perturbations Follow-up studies by J. Steinberger (1949) made use of the then new technique of Feynman diagrams to effect the perturbative calculation. 2 Steinberger represented neutral pion decay by the graph in Fig. 2, where the central triangular ‘loop’ corresponds to a linearly divergent integral: ‘All infinities in field theory are similar to that of this example. Somewhere in the Feynman diagram there is a closed loop which gives rise to the infinite integral’ (p. 1181). To deal with this Steinberger borrowed a regularisation technique (Pauli-Villars regularisation) and wrote down general consistency conditions for a formal ‘method of subtraction fields’ (p. 1180), applying it first to the no - yy decay. Two years later Schwinger (1951) treated the same problem as an application of his general ‘proper-time method’ (p. 672). Using Green’s functions without the aid of the Feynman diagrams. Schwinger’s treatment depended on formulating the perturbation expansion in such a way that a so-called proper- time integration was reserved to the last. Both Schwinger and Steinberger based their perturbation calculations for the neutral pion decay rate on approximations to an explicit Lagrangian function with interaction g(vy5 ~4 (where q/, II/ are the proton fields, r#~is the pion field andg is the coupling constant.) The concordance of these methods for handling the divergences suggested that something was right about the renormalisation techniques. But clearly some things, if not actually wrong, were certainly not well understood. 3 The problem of neutral pion decay arose again in the context of the ‘current algebra’ approach to symmetry introduced by M. Gell-Mann (1962) and the exploration of the assumption (called PCAC) that axial-vector currents are partially conserved: explicitly, that the divergence of the current is proportional to the particle mass. Arguments by D. G. Sutherland (1967) and M. Veltman (1967) implied that under this assumption (given gauge invariance) the am- 2 Steinberger’s paper appeared in the October issue of Physical Review. The September issue contained Feynman’s (1949) now famous paper, ‘Space-Time Approach to Quantum Electrodynamics’. 3 Concerning the cogency of Steinberg& calculation R. Jackiw remarked: ‘It is presumably an accident that this completely implausible calculation gives a result in excellent agreement with experiment’ (1972, p. 167). 310 Studies in History and Philosophy of Modern Physics plitude for rro - yy vanishes in the limit where the pion mass goes to zero. In this limit exact chiral symmetry holds, and J. C. Taylor (1958) had already shown that chiral symmetry is incompatible with the beta decay of pions. J. S. Bell and R. Jackiw (1969) formulated the problem this way: perturbation theory calculations involving methods for subtracting infinities yield a non- zero amplitude for neutral pion decay in the limit where the pion mass is zero. PCAC, however, which is formally satisfied in the Lagrangian models on which the perturbative calculations are based, implies that the decay amplitude vanishes in that limit. How can we reconcile the two? The answer provided by Bell and Jackiw was to suggest an unorthodox regularisation scheme. Their approach modifies the customary Pauli-Villars regularization. Bell and Jackiw require that the coupling constants gi for the auxiliary fermion fields used to ‘subtract off’ the infinities in the triangle diagram co-vary with the regulator masses mi of those fields; i.e. that mi/gi = m/g = constant (where g is the original coupling constant and m the fermion mass). This scheme preserves gauge invariance and the PCAC conditions (along with Lorentz covariance and Bose symmetry). The price for preserving these symmetries in the regularisation is the Sutherland-Veltman theorem: the neutral pion decay rate vanishes. In addition to this awkward conflict with experiment, S. Adler (1969) points out that for strong interactions the Bell and Jackiw regularisation would result in unrenormalisable infinities. Adler concludes that in order to preserve the other symmetries and also the possibility of renormalisation the PCAC condition must go. Adler’s analysis pinpoints the problem. The PCAC condition arises by calculating the divergence of the axial-vector current formally from the equations of motion inherent in the Lagrangian of the interaction model.